On the other hand, \(M\)  and \(-Q - \mu I_Y\)  are both maximally monotone and \(D(-Q - \mu I_Y) = Y\) . From Fact REF , \(M -Q - \mu I_Y\)  is maximally monotone, and Fact REF  provides \(y \in Y\)  such that \(0 \in \mu y + (M -Q - \mu I_Y)y\) , that is to say, \(0 \in (M - Q)y\) . It follows easily from this that \(G(M) \cap G(Q) \ne \emptyset \) , which completes the proof of Theorem REF .

We conclude this section with applications to bounded linear operators. Our analysis depends on the following result about monotone functions. See[1]}
